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Proportional Reasoning

Proportional Reasoning. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm , candle Q had burned 10 mm . When candle Q had burned 35 mm , how many mm would candle P have burned?.

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Proportional Reasoning

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  1. Proportional Reasoning Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? • Strategy 1: Setting up a proportion • Strategy 2: Coordinating quantities 0 mm 56 mm 16 mm 48 mm 32 mm 8 mm 16 mm 16 mm 16 mm 30 mm 10 mm 35 mm 0 mm 20 mm 10 mm 10 mm 10 mm Which strategy demonstrates a conceptual understanding of proportional relationship? 5 mm

  2. Objective: Making Connections Proportion a x b d = What is a proportion?

  3. Objective: Making Connections Proportion Beginning Algebra y = mx + b a x b d = y = mx How can we link these two ideas?

  4. Key Ideas • Focus on Co-variation and Invariance • Identifying quantities thatchange(i.e. variables) Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? (a) Identify the quantities in this problem. These are values of quantities, not quantities! These are numbers, not quantities! mm 16 10 mm ? 35 mm The length burned for candle P at the first moment. (given) The length burned for candle P at the second moment. (unknown) The length burned for candle Q at the first moment. (given) The length burned for candle Q at the second moment. (given)

  5. Key Ideas • Focus on Co-variation and Invariance • Identifying quantities thatchange(i.e. variables) and how those quantities are related Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? (a) Identify the quantities in this problem. (b) Let p represent the number of mm that candle P had burned when candle Q had burned q mm. Write an equation to relate p and q. Mentally act out the problem situation. Draw diagrams to represent the problem situation.

  6. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? 10 mm 16 mm • Which candle is skinner? • Candle P • Candle Q • The same P Q 1st moment

  7. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? 10 mm 16 mm • Which candle is skinner? • Candle P • Candle Q • The same P Q 1st moment

  8. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? 10 mm 16 mm 35 mm ? P Q Q P 2nd moment 1st moment

  9. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What is invariant in this problem? The burning rate of each candle. 10 mm 16 mm 35 mm ? P Q Q P 2nd moment 1st moment

  10. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What is invariant in this problem? The burning rate of each candle. 10 mm 16 mm 35 mm ? P P Q Q Q P 2nd moment 1st moment Initially

  11. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What is invariant in this problem? The burning rate of each candle. 2nd moment 1st moment Initially

  12. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What is invariant in this problem? The burning rate of each candle. P Q Initially

  13. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What is invariant in this problem? The burning rate of each candle. 2nd moment Initially

  14. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What else is invariant in this problem? The ratio of 16/10 is invariant. What does the ratio 16/10, or the value 1.6, represent? P Q Q P 2nd moment Initially

  15. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What else is invariant in this problem? The ratio of 16/10 is invariant. What does the ratio 16/10, or the value 1.6, represent?

  16. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What else is invariant in this problem? The ratio of 16/10 is invariant. What does the ratio 16/10, or the value 1.6, represent? Candle P burned 1.6 mm for every 1mm burned by Candle Q.

  17. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? What else is invariant in this problem? The ratio of 16/10 is invariant. What does the ratio 16/10, or the value 1.6, represent? Candle P burned 1.6 mm for every 1mm burned by Candle Q. p q (b) Let p represent the number of mm that candle P had burned when candle Q had burned q mm. Write an equation to relate p and q. q = p 1.6

  18. Key Ideas • Focus on Co-variation and Invariance • Focusing on quantities and relationships • Making connections among various representations

  19. Two different candles, P and Q, lighted at the same time were burning at different, but constant, rates. When candle P had burned 16 mm, candle Q had burned 10 mm. When candle Q had burned 35 mm, how many mm would candle P have burned? (c) How else can we show the relationship between the variables? Length burned (mm) x Candle P 35 Candle Q 16 10 Time 1st Moment 2nd Moment

  20. Key Ideas • Focus on Co-variation and Invariance • Focusing on quantities and relationships • Making connections among various representations • Interpreting slope meaningfully

  21. What does the slope represent of each line represent? & The burning rate for each candle (value is unknown). = How are the slopes related? Candle P burned 1.6 times as fast as Candle Q. Slope of line for Candle P is 1.6 times that of Candle Q. mP = 1.6mQ Length burned (mm) x 16 T1 Candle P 16 1.6 10 1 10 T1 16 T1 = 10 T1 35 Candle Q 16 10 16 10 Time 1st Moment 2nd Moment T1

  22. Key Ideas • Focus on Co-variation and Invariance • Focusing on quantities and relationships • Making connections among various representations • Interpreting slope meaningfully 16 x 10 35 = • Recognizing that ratio is invariant 16 p 10 q =

  23. p = 1.6q Length burned (mm) Candle P x x 35 p q 16 x 10 35 16 1.6 10 1 = = = = Candle Q 35 p x p q q 35 16 10 16 10 Time 1st Moment 2nd Moment

  24. Key Ideas • Focus on Co-variation and Invariance • Focusing on quantities and relationships • Making connections among various representations • Interpreting slope meaningfully 16 x 10 35 35 x 10 16 = = • Recognizing that ratio is invariant • Relating the meaning of ratio to the context of the problem

  25. We can solve this problem by setting up a proportion like . 35 x 10 16 = The ratio 35/10 is equal to 3.5. What is the significance of 3.5 in terms of the burning candles? Length burned (mm) ? Candle P 35 30 Candle Q 20 16 10 Time 1st Moment 2nd Moment

  26. We can solve this problem by setting up a proportion like . 35 x 10 16 = The ratio 35/10 is equal to 3.5. What is the significance of 3.5 in terms of the burning candles? Length burned (mm) ? Candle P 48 35 32 Candle Q 16 10 Time 1st Moment 2nd Moment

  27. Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm. At what time were the two candles lighted? b. What is the burning rate for candle Q? c. Suppose the original length of candles P and Q are 200 mm. Which candle is skinnier? Length burned (mm) 200 35 16 10 Time 7:40pm 8:00pm 8:50pm ?

  28. Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm. Let t be the # of minutes since the lighting of the candles. Let bP be the length of candle P that has burned at time t. Let bQ be the length of candle Q that has burned at time t. (mm) bP vs t graph 200 bQ vs t graph 35 16 10 t (min) 0 20 70

  29. Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm. Let t be the # of minutes since the lighting of the candles. Let bP be the length of candle P that has burned at time t. Let bQ be the length of candle Q that has burned at time t. Let hP be the height in mm of candle P at time t. (mm) bP vs t graph 200 bQ vs t graph hP vs t graph t (min) 0

  30. Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm. Let t be the # of minutes since the lighting of the candles. Let bP be the length of candle P that has burned at time t. Let bQ be the length of candle Q that has burned at time t. Let hP be the height in mm of candle P at time t. Let hQ be the height in mm of candle Q at time t. (mm) bP vs t graph 200 bQ vs t graph hQ vs t graph hP vs t graph t (min) 0

  31. Two different candles, P and Q, lighted at the same time were burning at different but constant rates. At 8:00pm candle P had burned 16 mm and candle Q had burned 10 mm. At 8:50pm candle Q had burned 35 mm. Write an equation to relate the variables in each pair and briefly describe the meaning of the slope and y-intercept of your equation. (i) bP and t (ii) bQ and t (iii) hP and t(iv) hQ and t (v) hP and bP (vi) bP and bQ (mm) bP vs t graph 200 bQ vs t graph hQ vs t graph hP vs t graph t (min) 0

  32. All the contextualized problems in this presentation are from this article in Mathematics Teaching in the Middle School Vol. 14, No. 8, April 2009

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